So you either need every app developer to maintain the list of all push providers or someone, presumably Google, to maintain the canonical list and the thing that manages sending them to all the push providers.
Oh and then there's the associated suite of privacy issues. Do you really want every push notification prouder to get even metadata about all of your push notifications?
So this can't be a global setting. Instead it has to be a per-app setting where like the app provider needs to register a callback to update the notification server and support that in app. Of course most won't.